Cosmological Model in $f(R, \mathcal{G})$ Gravity : Stability Analysis and Observational Constraints from DESI DR2

In this article, we examine the dynamical system of the Decoupled Power-law $f(R,\mathcal{G})$ gravity model. This $f(R,G)$ model framework is composed of interactions between dark matter and scalar field through the linear coupling term. The key objective of the present study is to describe the cosmological viability of the modified gravity theory formulated with gravity $ f(R, \mathcal{G}) $. We transform the cosmological equations into an autonomous system of ordinary differential equations by suitable transformation of variables. The decoupled power-law $ f(R,\mathcal{G})$ model governed by $ f(R, \mathcal{G}) = \alpha R^m + \beta \mathcal{G}^n $ has been investigated in detail to characterize the stability properties of the critical points of the autonomous system. The model may explain the late-time accelerating universe expansion corresponding to the attractor in the model. Depending on the effective equation of state parameter values corresponding to the critical points, we study the observational viability of the model using low-redshift observational data, such as observational Hubble data. Furthermore, we investigate the effects of parameters using the effective equation of the state parameter and the statefinder diagnostics. We further investigate the observational viability of the model by constraining its parameters through Markov Chain Monte Carlo (MCMC) analysis of the combined cosmic chronometer, Pantheon+SH0ES, CMB, and DESI DR2 BAO data. The constrained model predicts $H_0 = 69.71 \pm 0.61~\mathrm{km\,s^{-1}\,Mpc^{-1}}$, $q_0=-0.530$ with a transition redshift $z_t=0.646$, and a quintessence-like effective equation of state. The cosmographic parameters and cosmic age are also consistent with $\Lambda$CDM expectations.